Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence
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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 9, SEPTEMBER 1997 1735
Application of the Manakov-PMD Equation
to Studies of Signal Propagation in Optical
Fibers with Randomly Varying Birefringence
Dietrich Marcuse, Fellow, IEEE, C. R. Menyuk, Senior Member, IEEE,
and P. K. A. Wai, Senior Member, IEEE
Abstract— We report here on our investigations of the varying envelope approximation applies, which is the case in
Manakov-polarization mode dispersion (PMD) equation which communication signals to date [2], and fiber birefringence
can be used to model both nonreturn-to-zero (NRZ) and soliton
can be ignored, then the transmission is well-described by
signal propagation in optical fibers with randomly varying
birefringence. We review the derivation of the Manakov-PMD the nonlinear Schrödinger equation. Unfortunately, it is of-
equation from the coupled nonlinear Schrödinger equation, and ten impossible to ignore the fiber birefringence, particularly
we discuss the physical meaning of its terms. We discuss our in modern-day NRZ (nonreturn-to-zero) communications, in
numerical approach for solving this equation, and we apply this which polarization scrambling is consistently used. Menyuk
approach to both NRZ and soliton propagation. We show by
comparison with the coupled nonlinear Schrödinger equation, [3] showed that the CNLS (coupled nonlinear Schrödinger
integrated with steps that are short enough to follow the detailed equation) applies to birefringent optical fibers, but early work
polarization evolution, that our approach is orders of magnitude focused on fibers with constant birefringence [4]. In the pa-
faster with no loss of accuracy. Finally, we compare our approach rameter regime in which present-day communication systems
to the widely used coarse-step method and demonstrate that the
coarse-step method is both efficient and valid.
operate, the birefringence is large in the sense that the beat
lengths are typically 10–100 m, while the dispersive and
Index Terms—Birefringence, optical fibers, polarizations, ran- nonlinear scale lengths are hundreds of kilometers—but is
dom bifringence, solitons.
rapidly varying in the sense that the length scale on which
the birefringence orientation varies is 0.3 to 300 m. When
I. INTRODUCTION nonlinearity can be ignored, Poole and Wagner [5]–[7] showed
that the large but rapidly and randomly varying birefringence
B ECAUSE of their low loss, optical fibers have become
an important medium for long-distance communications.
Even though the glass fiber is a remarkably linear transmission
leads to PMD (polarization mode dispersion), which is a
very important effect in communication fibers. An important
medium, its slight nonlinearity, due primarily to the Kerr advance in dealing with both nonlinearity and the rapidly
effect, became important with the advent of the erbium- varying birefringence was made when it was recognized that if
doped optical amplifier that permitted transmission of light the polarization states of the light is averaged over the Poincaré
signals through optical fibers over transoceanic distances with- sphere, the Manakov equation can be derived from the CNLS
out regeneration, allowing the effect of the nonlinearity to [8], [9]. However, the assumption that the polarization states
accumulate over the entire distance. are mixed on the Poincaré sphere was not justified, and the
When light travels in an optical fiber for thousands of nature of the correction terms was not completely elucidated,
kilometers, the effect of the nonlinearity can no longer be although it was recognized that they are important [8]; it
ignored. On the one hand, it can lead to significant sig- is these terms that are responsible for PMD. Since then,
nal distortion—particularly when coupled with the sponta- in a series of works that carefully investigated the statistics
neous emission noise due to the erbium-doped fiber amplifiers on the Poincaré sphere using physically reasonable models
(EDFA’s); on the other hand, it can be used to counteract [10], [11], we were able to justify the assumption in [8]
dispersion, as in solitons. The development of communications and [9] and to elucidate the nature of the corrections to the
technology has made it necessary to develop mathematical Manakov equation. This work culminated in the derivation of
tools for describing light transmission in nonlinear optical the Manakov-PMD equation [11] which it is the object of this
fibers. Hasegawa and Tappert [1] showed that when the slowly paper to study.
It is our view that the derivation of the Manakov-PMD
Manuscript received November 8, 1996. This work was supported by ARPA equation completely resolves the outstanding theoretical issues
through the AFOSR, NSF, and DOE.
D. Marcuse and C. R. Menyuk are with the Department of Computer Sci-
concerning appropriate models for modeling birefringence in
ence and Electrical Engineering, University of Maryland, Baltimore County, optical fibers. Moreover, this equation leads to highly rapid
Baltimore, MD 21228 USA. numerical schemes and can be used to justify the widely
P. K. A. Wai is with the Department of Electronic Engineering, The Hong
Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. used coarse-step method. From a physical standpoint, the
Publisher Item Identifier S 0733-8724(97)06657-7. mathematical transformation that allows us to derive the
0733–8724/97$10.00 1997 IEEE1736 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 9, SEPTEMBER 1997
Manakov-PMD equation from the CNLS equation amounts to single guided mode which, however, can exist in two mutually
freezing the linear motion of the signal’s central frequency on orthogonal polarization [12]. In a perfectly isotropic, circularly
the Poincaré sphere. Since the motion of the other frequencies symmetric fiber the two polarization components of the
in the signal and the changes due to nonlinearity are slow mode would travel with identical phase and group velocities.
relative to this point’s motion on the sphere, we can take long Unavoidable fiber imperfections break this polarization
computational steps. The terms that appear in the Manakov- degeneracy so that at each point along the fiber there exists
PMD equation have a transparent physical meaning. There are a preferred direction perpendicular to its axis. A light field
two terms that account for the Kerr nonlinearity averaged over polarized orthogonal to this direction has the smallest phase
the Poincaré sphere and chromatic dispersion; these are the and group velocity. We designate the propagation constant of
terms that appear in the Manakov equation. There is another the wave polarized in the preferred direction as and the
term that corresponds to the usual linear PMD and when both propagation constant of the orthogonally polarized wave as .
chromatic dispersion and nonlinearity can be neglected, this The difference of these two propagation constants is defined
term yields all the effects that Poole and others have described. as the birefringence parameter . The specific
Finally, there are additional nonlinear terms that have not group delay per unit length is represented by
been previously discussed in detail (but see [11]) and that where the primes indicate differentiation with respect to the
we refer to as nonlinear PMD terms. We will show that these angular frequency. In this paper, in contrast to our earlier
terms, which are due physically to incomplete mixing on the paper [11], we do not use normalized units so that and
Poincaré sphere, are vanishingly small in cases of present- have the dimension of the inverse length, while and have
day, practical importance in optical communications and can dimensions of time divided by length. In this paper, we also
safely be ignored in these cases. use to designate the preferred direction, rather than
At the present time, most simulations of polarization effects as in [11] since is the geometric rotation angle. The angle
in long-distance optical fiber transmission are done using a rotates randomly; and may also vary randomly.
somewhat ad hoc procedure that is discussed in [8] and [9] and Optical signals in fibers can be distorted due to a combi-
that we refer to here as the coarse-step method. The starting nation of several effects. First, if , the signals undergo
point for this approach is the CNLS without the rapidly varying PMD [5]–[7]. Second, if the signals undergo
terms that is described in [3] and [4]. One then integrates chromatic dispersion. Finally, optical fibers are weakly nonlin-
these equations using step sizes that are large compared to ear which means that the refractive index of the fiber material
the fiber correlation length—typically several kilometers—and changes as a function of the light intensity. The propagation
then randomly rotates the data on the Poincaré sphere. This of light pulses, taking all these effects into account, can be
approach leads to an artificially increased fiber correlation described by the coupled nonlinear Schrödinger equation [3],
length, but one also artificially reduces the strength of the [4]
birefringence. By doing so, one can obtain the same magnitude
for the linear PMD as one would in simulations which use
realistic fiber correlation lengths and birefringence strengths,
and one realizes an enormous saving in computation time. We (1)
shall show in this paper that this approach, as it has been
used to date, both exaggerates the nonlinear PMD and deals or equivalently
with it inaccurately, but we will also show that the nonlinear
PMD is so small in cases of present-day practical importance
in long-distance communication systems that this exaggeration
does not matter. Moreover, we will show how to eliminate this (2)
difficulty. Thus, our work provides a theoretical justification
which are written in a form that is appropriate for a fiber
for the use of the coarse-step method.
with a randomly varying birefringence orientation. Here,
The reminder of this paper is organized as follows: In
is a column vector with elements and which
Section II, we review the derivation of the Manakov-PMD
are the complex envelopes of the two polarization components.
equation, and in Section III, we discuss the numerical algo-
The vector is defined as , where designates
rithms for solving it. In Section IV, we apply this equation
complex conjugation. The -coordinate measures distance
to examples of NRZ and soliton communication signals, and
along the fiber axis, while the -coordinate represents retarded
we discuss the nonlinear PMD. In Section V, we compare the
time—the time relative to the moving center of the signal.
Manakov-PMD equation to the coarse-step method. Section VI
The matrix
contains the conclusions.
(3)
II. DERIVATION OF THE MANAKOV-PMD EQUATION is defined in terms of the Pauli’s matrices
In this section, we will review the derivation of the I
Manakov-PMD equation that was earlier discussed in [11],
(4)
presenting it from a more physical perspective. Optical fibers
used in telecommunications are designed to support only aMARCUSE et al.: APPLICATION OF THE MANAKOV-PMD EQUATION 1737
and by using the same matrix in the second and third Second, we introduce a new field vector via the unitary
terms of (1), we are effectively assuming that the orientation matrix T
of the axes of birefringence is only a weak function of
frequency. The nonlinear Kerr coefficient is represented by T (9)
and the wavenumber by , where is the vacuum
wavelength of light. The parameter in (1) and (2) does not where T is the solution to the ordinary differential equation
have a subscript since we are assuming that . T
A fundamental difficulty in solving (1) or equivalently (2) T (10)
is that changes on a length scale of 0.3–100 m, while the
length scales for PMD, chromatic dispersion, and the Kerr with the initial condition
nonlinearity are hundreds or even thousands of kilometers. If
one wishes to solve (1) accurately, taking into account rapid T (11)
and random variations of the birefringence, one must do so
on a very short length which is computationally expensive. One then finds that (1) yields in the linear,
Moreover, the different physical effects that contribute to the CW limit, so that is frozen on the Poincaré sphere in this
signal evolution in (1) are mixed together in a complicated limit. In effect, the transformation T follows the rapid and
way. It is known that when the nonlinearity and chromatic dis- random motion of a single state on the Poincaré sphere. This
persion can be ignored, the signal undergoes polarization mode motion occurs on a length scale of 0.3–100 m. Of course,
dispersion [5]–[7]. Conversely, when the PMD is small but the is not frozen on the Poincaré sphere at all -points
birefringence is rapidly and randomly varying, it is known that except in the special case of a linear CW field. However, in
the Manakov equation applies. It is not immediately apparent all cases of present-day interest in communication systems,
where in (1) these effects are embedded. its evolution is quite slowly varying—on a length scale of
The Manakov-PMD equation resolves both these difficul- hundreds of kilometers, associated with the scale lengths of
ties. It not only allows us to develop a rapid numerical PMD, chromatic dispersion, and nonlinearity. This physical
scheme for determining the signal evolution with no loss observation is at the heart of our approach.
of accuracy, but the different physical effects are isolated in After substitution of this transformation into (1), we obtain
separate terms of the equation that have a transparent physical the Manakov-PMD equation [11]
meaning. Thus, this equation is the natural starting point for
analysis of signal propagation in optical fibers. In particular,
we will use it to validate the widely used coarse-step method
in the parameter regime that is relevant to long-distance
communication systems. (12)
To obtain the Manakov-PMD, we proceed in two successive
steps [11]. First, the field is transformed from the fixed When the right-hand side is zero, (12) reduces to the Manakov
coordinate system to a coordinate system that rotates with the equation. The second term on the left-hand side of (12)
principal axes of the fiber. The new field is designated by the includes the effect of chromatic dispersion, and the third
vector that is related to as follows: term on the left-hand side of (12) includes the effect of Kerr
nonlinearity averaged over the Poincaré sphere with the well-
R known 8/9 factor [8], [9]. We will show that the first term on
the right-hand side of (12) leads to the usual linear PMD that is
(5) discussed in [5]–[7]. Finally, the second term on the right-hand
side of (12) is a new term that we refer to as nonlinear PMD.
If we consider a continuous wave (CW) signal at a single Physically, it is due to incomplete mixing on the Poincaré
frequency and neglect the fiber nonlinearity, then (1) reduces sphere. To our knowledge, this effect has not been previously
to discussed except briefly near the end of [11]. In this paper, we
will show that its effect is negligible in the parameter regime
in which communication fibers normally operate.
(6) There are three quantities, , and , that require
special discussion. The matrix T defined in (9) and (10) is
and the transformation (5) implies [13] a special unitary matrix so that it has the form
(7) T (13)
where, letting , we find with the additional constraint . We note that
our notation differs from that of [11] in which the vector
(8) now corresponds to the continuous wave
signal of (7). The other solution of (7) is given by1738 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 9, SEPTEMBER 1997
. We define the Stokes parameters of the CW wave III. NUMERICAL APPROACH
as The numerical approach that we follow in solving the
Manakov-PMD equation is closely related to the widely
used split-step operator technique that is used to solved the
nonlinear Schrödinger equation and the coupled nonlinear
(14) Schrödinger equation and is discussed, for example, in [2]
and [4]. In this approach, time derivatives are evaluated
Using the solutions and , we also define in the frequency domain while functional multiplications
are evaluated in the time domain. For the usual nonlinear
Schrödinger equation and the coupled nonlinear Schrödinger
equation, which do not take into account such complications as
the Raman effect, this prescription implies that the linear terms
(15) are evaluated in the frequency domain while the nonlinear
terms are evaluated in the time domain. By splitting the spatial
Defining the real vectors and recalling that
step, so that the time domain evaluations are done in the middle
at and , it can be shown that
of the frequency domain evaluations, it is possible to maintain
the are vectors on the Poincaré sphere [11] and that is
second-order accuracy with one function evaluation per step
initially pointed in the -direction, is initially pointed in
for each term at every node point. We also note, because it is
the -direction, and is initially pointed in the -direction.
important in practice, that certain linear effects like chromatic
As T evolves in accordance with (10), the will wander on
dispersion do not change the spectral intensity and certain
the surface of the Poincaré sphere but will remain orthogonal
nonlinear effects like the Kerr nonlinearity do not change the
to each other. We have changed notation from that of [11] to
emphasize the geometric connection to vectors on the Poincaré temporal intensity. In order to avoid numerical instabilities, it
sphere. We now find that is important that the numerical approximations of these effects
retain these properties to far higher than second order, even
though the overall scheme is only second-order accurate.
T T (16)
The vectors and are related to each other by a simple
unitary transformation which can be found by integrating
We also find that where (10) and is rapidly determined, since it is only a second-
order ordinary differential equation, even though it must
be integrated on the short length scale given by the beat
length and fiber correlation length. In most practical cases,
it is not even necessary to transform from one to the other.
Since the electrical output current from a realistic detection
(17) system is given by , it does not matter which
we use. We stress this point because in devising numerical
and algorithms for solving (12) efficiently, we have found it
somewhat advantageous to work with (1) rather than (12).
The linear part of (1) may be written in the form
(18)
(19)
The Manakov-PMD equation is useful from a conceptual
standpoint because each of the principal effects that contribute which in the Fourier domain becomes
to the signal evolution, the Kerr effect, chromatic dispersion,
and polarization mode dispersion are clearly separated. The (20)
Manakov-PMD equation is useful from a numerical standpoint
because is slowly varying. The and that appear
The solution to (20) may be written
in this equation are rapidly varying, but they are determined
by solving (10) which is merely a second-order ordinary
M (21)
differential equation and can be rapidly solved. When solving
Eq. (12) on the slowly varying scale, one must properly
where the transfer matrix M [14] satisfies the ordinary differ-
average over the rapidly varying terms, i.e., one must replace
ential equation
the by , and we will discuss this issue
in the next section. In principle, it is possible to take advantage M
of the central limit theorem to replace these averages with M (22)
appropriate random variables [11]. This issue is currently being
studied. with the initial condition M I, where I is theMARCUSE et al.: APPLICATION OF THE MANAKOV-PMD EQUATION 1739
2 2 identity matrix. To calculate M , we subdivided
each interval into smaller subintervals , and in each
subinterval we set
m (23)
where
(a)
(24)
(25)
and the are constant in each subinterval. The total transfer
matrix is now expressed at
M m (26)
(b)
The transfer matrix includes the rapid motion over the Fig. 1. Frequency dependence of the elements of the M-matrix for the CNLS;
(a) real and imaginary parts of M11 and (b) real and imaginary parts of M12 .
Poincaré sphere that occurs at the central frequency, but the
variation as a function of frequency will be quite small as long
as the interval is small compared to the PMD length scale. change in that occurs at every intervals , are given by
One way to reach this conclusion is to note, using (5), (9),
and (21), that
(28)
T R
M R T (27)
In Fig. 1(a) and (b), we show M and M ,
the real and imaginary parts of M , and M and
Since T and R are independent of , and the change in
M , the real and imaginary parts of M . The
is small over the interval M must vary slowly smooth, slow oscillations might seem surprising at first, con-
as a function of . Of course, we could directly calculate sidering that the fluctuate wildly over the 600 short fiber
M T R MRT, but this calculation is, in fact, slightly segments that makes up . However, we note that
more elaborate because depends on which has -function when GHz, which is larger than the
jumps. frequency range of interest even though the PMD is large
The transfer matrix M can be used to determine the compared to standard fiber parameters.
evolution of according to (21), or equivalently, we Dealing with the nonlinear terms in the time domain was
can use M to determine . If , as quite straightforward. The Manakov term is easily integrable,
was typically the case in the examples that we chose, and and we used the expression
we calculated M at each of the frequencies that we used
in the fast Fourier transform, then our procedure would still
(29)
be slow, although faster than direct integration of (1) using
the split step method by more than an order of magnitude.
However, the variation of M as a function of frequency to calculate its effect. To calculate the effect of the nonlinear
PMD terms, we integrated over the rapidly varying coefficients
is very slow on the length scale given by , and it is only
and then did a one-step Euler iteration of .
necessary to evaluate M at a few frequencies (typically 16)
and to interpolate between them with a second-order Lagrange
interpolation polynomial. IV. APPLICATIONS
To demonstrate the slow variation of M with , we Here, we will describe the application of the Manakov-PMD
consider an example in which km, m, equation to studies of both NRZ and soliton signal propagation.
m, ps/(km) , and We emphasize that since the Manakov-PMD is the basic
m, where is the fiber correlation length [10], [11], equation that describes pulse propagation in optical fibers,
is the beat length, and is the polarization mode taking into account the Kerr effect, chromatic dispersion, and
dispersion coefficient. We recall that the parameters , , and polarization mode dispersion, it is equally applicable to both
where is the standard deviation of the propagation formats. We will show that solving the Manakov-1740 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 9, SEPTEMBER 1997
(a)
Fig. 2. Dispersion map showing the cumulative dispersion as a function of
the length coordinate z .
PMD equation using the numerical scheme described in the
last section leads to precisely the same result as is obtained by
solving the coupled nonlinear Schrödinger equation on a short
(b)
length scale, although it is computationally about three orders
Fig. 3. The train of pseudorandom input pulses consisting of 16 bits. (a)
of magnitude faster in the cases that we considered. Finally, Power of the optical pulses in mW and (b) detected and filtered current in
we will discuss the nonlinear PMD term, and we will show mW units.
that it is negligible in communication fibers using present-day
parameters.
pulse power is mW. This power is somewhat high
for communications purposes but was chosen here to produce
A. NRZ Signals a sizable nonlinear effect in 500 km of fiber. Subsequently,
In our studies of NRZ signals we selected the following we present the shape of the received pulses as they appear
parameters in addition to those that we listed in the previous after square-law detection and filtering with a
section: the wavelength, m, the effective mode area, GHz “electrical” Bessel filter. This process distorts the pulses
m2 , the Kerr coefficient, m W, very slightly as can be seen in Fig. 3(b) which represents the
an FFT with grid points, and a total fiber length, square-law-detected, filtered input pulse. The quantity plotted
km. in Fig. 3(b) corresponds to a current so that it can go negative,
A very important point is the choice of a dispersion map. but it is here labeled in units of mW for comparison with
Since at zero dispersion the frequency spectrum of the signal the optical pulse, since detected current corresponds to optical
broadens uncontrollably [15], it is essential to have some power.
dispersion in the system. Using a fixed amount of dispersion All simulations reported here were run without fiber losses,
is undesirable since it causes linear pulse distortion which, on implying that there are also no optical amplifiers in the system
the one hand, is of little theoretical interest and on the other is and, consequently, no amplifier (ASE) noise.
undesirable in any realistic communication systems. For this To compute the behavior of the system, two approaches
reason, we use here a dispersion map consisting of a periodic were used. The first solved the coupled nonlinear Schrödinger
sequence of normal, , and anomalous, , fiber equation on the short length scale, carrying out a full split-
sections. The period length is 100 km, consisting of 80 km step calculation after traversing each short interval. This
sections with normal dispersion, ps/(nm km), and of program is extremely slow and requires 1.5 days of running
20 km compensation sections with ps/(nm km). Thus, time for the job at hand on a Sun SPARCstation 10. The second
at the end of each period and of course also at the end of program utilized the ideas explained in the preceding section
the fiber, the cumulative linear dispersion is zero. The shape and solved the Manakov-PMD equation in two minutes, and
of the cumulative dispersion map is shown in Fig. 2. In all we ran it both with and without the nonlinear PMD correction.
simulations reported here, the third-order frequency derivative The result of the simulation is shown in Fig. 4(a), which
of (and of course all derivatives of higher orders) were represents the “electrical” output pulses. Both approaches
zero. produced exactly the same output. Moreover, there is no
As the input pulse pattern we used a 16 bit pseudorandom discernible difference whether the Manakov routine is run with
word in NRZ (nonreturn to zero) format with a 0.2 ns pulse or without nonlinear PMD.
width corresponding to a signaling rate of 5 Gb/s. These The shape of the pulses, shown in Fig. 4(a), is strongly
pulses are generated by passing rectangularly shaped pulses dependent on the random number sequence used for computing
through a low-pass Bessel filter of GHz width, the random variations of the fiber orientation angle, . The
which gives the pulses slightly rounded edges. These optical overshoot seen at the trailing edges of the pulses in Fig. 4(a)
input pulses are shown in Fig. 3(a) in units of mW. The peak can be somewhat larger or smaller. They can even switch sidesMARCUSE et al.: APPLICATION OF THE MANAKOV-PMD EQUATION 1741
is caused by the interaction of the linear PMD with the
fiber nonlinearity. However, this interaction is almost entirely
due to the interaction of the linear PMD with the Manakov
nonlinearity, i.e., the Kerr nonlinearity averaged over the
Poincaré sphere, since inclusion of nonlinear PMD leads to
no “observable” effect.
B. Solitons
(a)
It is instructive to demonstrate by means of a numerical
example how fiber birefringence affects the soliton solutions
of the CNLS. Averaging over the fiber birefringence, which
is explicitly incorporated into the Manakov-PMD equation,
manifests itself by the appearance of the factor 8/9 on its left-
hand side. The Manakov-PMD equation is exactly equivalent
to the CNLS equation, but the averaged contribution of the
spatially varying birefringence, which appears on the left-hand
(b) side of the equation, has been explicitly separated from the
fluctuating contributions, which are on the right-hand side. The
nonlinear contribution on the right-hand side may be neglected
in most cases of practical interest. However, it is apparent
that in the limit that the birefringence strength tends to zero,
the right-hand side of the Manakov-PMD equation cannot
be negligible and must contribute an amount appropriate to
change the 8/9-factor to unity. Thus, fiber birefringence has
a noticeable influence on soliton solutions of the Manakov
equation where the 8/9-factor appears explicitly in the formula
for the soliton pulse duration which, in soliton units, assumes
(c) the form
Fig. 4. Detected and filtered current of the output pulses for Lcorr = 100 m
and beat = 50 m. (a) This figure shows the solution of the CNLS equation (30)
and the Manakov-PMD equation both with and without the nonlinear PMD
term. All figures look alike. (b) Same as (a) but for a linear fiber with n2 = 0.
(c) Same as (a) but with PMD = 0. To demonstrate the influence of fiber birefringence, the solid
curve in Fig. 5(a) shows the soliton solution of the Manakov
and appear at the leading edges of the pulses. These differences equation, computed without the negligible correction terms,
correspond physically to the observation that different fibers while the dotted curve represents the soliton solution of
with different random variations of the birefringence will the CNLS equation for a fiber without birefringence. We
have somewhat different behaviors. Nonetheless, our two have confirmed by numerical solutions of the differential
equations that both pulses maintain their shape as they travel
approaches produce exactly the same result for the same
along the fiber. Fig. 5(b) shows the soliton solution of the
assumed variation of the fiber birefringence.
CNLS for a fiber with birefringence. Superimposed on this
To identify the source of the pulse distortion, we “turn
curve is the Manakov solution of Fig. 5(a). Both curves are
off” the nonlinearity by setting , obtaining Fig. 4(b).
almost indistinguishable. However, the height of the CNLS
Since in a linear system the compensated dispersion map
soliton fluctuates along the fiber due to the randomness of the
prevents dispersion from distorting the pulses, the remaining
birefringence. The maximum and minimum excursions of the
slight pulse distortion seen in Fig. 4(b) is caused entirely by
calculated pulse heights differ by approximately 1% of the
linear PMD which is not affected by the dispersion map.
average pulse height, but Fig. 5(b) demonstrates clearly that
However, not all the distortion in Fig. 4(a) is due to the
the fiber birefringence does indeed modify the shape of the
interaction of PMD and the fiber nonlinearity, as can be seen
soliton pulse by the amount predicted by the 8/9-factor in the
in Fig. 4(c), which was computed with the regular value of
Manakov equation.
but with . This figure shows that the nonlinearity
conspires with dispersion to distort the pulses even though
the overall cumulative linear dispersion of the system is zero. C. Nonlinear PMD
The slight asymmetry of the pulses in Fig. 4(c) is attributable Mathematically, the terms that we refer to as nonlinear
to the “electrical” filter which shows this tendency already in PMD corresponds to the difference between the Kerr effect
Fig. 3(b), but which is enhanced by the substantial broadening terms and their average over the Poincaré sphere. Physically
of the spectrum at the end of the fiber. A comparison of these terms correspond to the rapidly varying fluctuations
Fig. 4(a)–(c) makes it clear that the bulk of the pulse distortion in the strength of the Kerr effect as the polarization state1742 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 9, SEPTEMBER 1997
varying birefringence and, in a fixed frame, the electric field
remains correlated over long lengths. This increased mixing
length only affects the nonlinear PMD; the strength of the
linear PMD is proportional to . It has been proposed to
significantly decrease the linear PMD by rapidly shifting the
orientation of the birefringence axes in a fiber as it is drawn,
which amounts to reducing while keeping fixed
[11]. This approach raises the specter that nonlinear PMD
(a) might become important in these fibers, but that is not the
case. An analytical calculation of the mixing length in this
limit shows that it equals [11], and the
factor of is so small that it is difficult to achieve a
significant mixing length even under extreme circumstances.
To illustrate this point, we carried out simulations with
the same parameter set as in Section IV-A, except that we
changed from 100 to 10 m and from 50 m to
10 km. Obviously, a beat length of 10 km is far larger than
can be presently achieved in practice. Nonetheless, the mixing
length is only 100 km in this case, which while approaching
(b) the nonlinear and dispersive scale lengths is not larger than
Fig. 5. (a) The solid curve is the solution of the Manakov equation, i.e., them. Even under these fairly extreme conditions, we found
the Manakov-PMD equation without the PMD terms on its right-hand side, no significant effect from the nonlinear PMD terms. To observe
while the dotted curve is the soliton solution of the CNLS equation for a
fiber without birefringence. (b) Soliton solution of the CNLS equation for a the effect of the nonlinear PMD terms, we had to further lower
fiber with birefringence. Superimposed on this curve is the Manakov solution from 10 m to 1 m. In this case, the mixing length equals
of (a). The height of the CNLS soliton fluctuates by about 1% due to the 1000 km which is larger than the nonlinear and dispersive
randomness of the fiber birefringence.
scale lengths. (Note that in this case, we set
instead of .)
changes. The size of these fluctuations does not depend on The pulse shape computed for these conditions with the
the birefringence strength; they are about the same size as the CNLS as well as with the Manakov-PMD equation, including
Kerr effect itself. However, their duration depends on the rate the nonlinear correction term, is shown in Fig. 6(a). The
at which the polarization state visits the different points on the figures computed with the two programs are indistinguishable.
Poincaré sphere. If this rate is rapid—as is the case in standard Fig. 6(b) shows what happens when we neglect the nonlinear
communication fibers with standard signal parameters—then PMD. The pulse shape in Fig. 6(b) is distinctly different
the fluctuations rapidly change sign and their net effect on the from that of Fig. 6(a). This example clearly shows that the
signal evolution is nearly zero. nonlinear correction term in (12) does make a contribution
A significant contribution from nonlinear PMD can only if the inequality is sufficiently well satisfied.
occur when mixing on the Poincaré sphere is poor, i.e., this However, this example represents a highly extreme case that is
mixing occurs on the same length scale or even a longer unlikely to be found in practice so that the nonlinear correction
length scale than is associated with the Kerr effect, chromatic term can safely be neglected for all communications fibers in
dispersion, and polarization mode dispersion. From the studies use today.
in [10] and [11], one concludes that there are two limits in
which this situation can occur. The first and most obvious is
when , the fiber correlation length, becomes comparable V. COMPARISON WITH THE COARSE-STEP METHOD
to the nonlinear and dispersive scale lengths. Since the mixing From a practical standpoint, it is not feasible to integrate
length on the Poincaré sphere must be longer than , the coupled nonlinear Schrödinger equation on the small
it is apparent that mixing will be poor. This limit, which length scale that is required to mimic the rapidly changing
applies to subpicosecond pulses, does not occur in present-day birefringence orientation. In the absence of the approach based
communication systems. In this limit, there is no advantage on the Manakov-PMD equation that we have described here,
to solving the Manakov-PMD equation rather than directly an ad hoc approach that was first introduced in [8] and [9]
solving the coupled nonlinear Schrödinger equation since there has become widely used. We refer to it here as the coarse-step
is no short length scale any more. method, and we will analyze it using the theoretical tools that
The second limit is a little less obvious. When were developed in Section II. We will show that the coarse-
, the mixing length becomes proportional to , step method as it is presently used deals with the nonlinear
and when the mixing length approaches the nonlinear and PMD inaccurately but that this inaccuracy does not matter
dispersive scale lengths, mixing will be poor [10], [11]. In when the contribution of the nonlinear PMD is negligible as
this limit, which corresponds to decreasing the magnitude of is the case in present-day communication systems. We will
the birefringence, the birefringence becomes sufficiently small show that it deals with the linear PMD accurately but to do so
that the polarization state is nearly unaffected by the randomly the magnitude of the birefringence must be artificially lowered.MARCUSE et al.: APPLICATION OF THE MANAKOV-PMD EQUATION 1743
where we have transformed from to
and then removed the primes. In the coarse-step method used
in [8] and [9], one simply solves (33) on a length scale that is
somewhat longer than the fiber correlation length—100 m was
used in [8] and 1 km in [9]—but that is still short compared to
the nonlinear and dispersive scale lengths.1 Since the random
variations of the birefringence are no longer present in (33),
the scattering of the field on the Poincaré sphere that they
(a) induce must be put back. In both [8] and [9], the authors do
that by scattering the field on the Poincaré sphere at the end
of each step, setting S . In [8], the authors used the
scattering matrix
S (34)
while in [9], the authors used
S (35)
(b)
where and are randomly chosen from uniform
Fig. 6. Detected and filtered current of the output pulses for Lcorr = 10 distributions in the first case and and are randomly chosen
m and beat = 10 000 m. (a) This figure shows the solution of the CNLS
equation and the Manakov-PMD equation with nonlinear PMD. (b) This figure from uniform distributions in the second case. In neither case
shows the solution of the Manakov-PMD equation without nonlinear PMD. does this matrix introduce a uniform scattering on the Poincaré
sphere. To achieve that, one must use, for example, the three
Euler angles [16]. However, concatenating several of these
We will also show how the method can in principle be modified matrices together does lead to rapid uniform mixing on the
to deal accurately with nonlinear PMD. Thus our results serve Poincaré sphere, as we will show. This point is significant
to validate the use of the coarse-step method in simulations of because the Manakov-PMD equation is obtained by removing
current communications systems. the rapid motion that a linear CW signal undergoes on the
Our starting point is the coupled nonlinear Schrödinger Poincaré sphere. In this case, a linear, CW signal is unaffected
equation transformed into the frame of the local axes of by (33) and the transfer matrix T just consists of a series of
birefringence. Substituting , given by (5) into (1), we obtain S-matrices multiplied together.
To demonstrate the uniform mixing on the Poincaré sphere
and to determine the mixing rate, we define the three Stokes
parameters , and
which characterize the coordinates on the Poincaré sphere. For
(31) comparison, we note that , , and correspond respectively
to , , and in [11]. (The minus sign appears in the
which may also be written representation used in [11] for is to be consistent with the
definition used in [7] of the Stokes parameters. Here we return
to the definition used in [9] and [13].) For the scattering matrix
given in (34), one finds that the Stokes parameters of a linear,
CW signal evolve according to the formula shown in (36) at
(32) the bottom of the next page and for (35), this result becomes
where . In the case where the birefrin-
gence axes are fixed, i.e., in equals 0, it was long ago
observed [3], [4] that the term is rapidly varying and could
be dropped from (31) with no loss of accuracy in the parameter (37)
regime that is used in communication fibers. Equation (31)
then reduces to
where the subscript indicates the step number. Noting that
the and are chosen randomly in each step for the
1 In the description of the numerical algorithm in both [8] and [9], it was
(33) not explicitly stated thatN is dropped. We are grateful to Dr. Evangelides
for describing to us the numerical algorithm that was used in [9].1744 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 9, SEPTEMBER 1997
first case, one finds from (36) that where letting
, and at .
(38) Noting that F [13], we substitute (44)
into (43) and use the results, which may be derived directly
from (36) and (37), that the are uncorrelated from step to
where the angular brackets indicate an average over all possi- step to conclude
ble values of and . We finally conclude
(45)
(39)
By contrast, one finds in the large- limit from [11, eq. (22)]
where if and 0 otherwise, and the values (46)
of and depends on the initial conditions. For example, if
, , and , we find and . Thus, the polarization mode dispersion will be too large by a
The convergence to a uniform distribution function is very factor of .
rapid. Indeed, if we define the diffusion length as the length However, if one lowers and hence the birefringence,
over which the initial conditions fall to within of their by a factor of , then the polarization mode
uniform value of , then we find that this occurs dispersion returns to its original value in a statistical sense.
within two steps, regardless of the initial conditions. A similar If, for example, m and km, one must lower
analysis for (37) yields the birefringence by a factor of four. The birefringence is often
not even known within a factor of this magnitude!
We now consider the nonlinear terms. The first point to note
(40) is that , given in (31), does not contribute to the average over
the Poincaré sphere which is regardless of
The convergence in this case is even more rapid than in the whether the term is kept or dropped. By contrast, the effect
previous case. on the nonlinear PMD is profound. We previously found that
We now determine the polarization mode dispersion in the nonlinear contribution involves the , as can be seen by
the coarse-step method. Focusing on the linear part of (33), reference to (12)–(18). In this case, we find that the nonlinear
ignoring chromatic dispersion, and the Kerr effect, we find contribution to is given by
S S (47)
S S S S S S where , given by (16) involves the , rather than the .
S S (41) We can write out an expression that is analogous to (17) but
there is little point. The basic result is that the nonlinear PMD
where is the Fourier transform of , the S are
is inaccurately described by the coarse-step method. It was
the scattering matrices given by (34) or (35), and
already noted in [11] that the coarse-step method tends to
S exaggerate the strength of the PMD coefficients by a factor
of about where is the appropriate mixing
(42) length. Even more serious, however, is that the coefficients
Taking the derivative of (41) with respect to and writing for nonlinear PMD are incorrect in the coarse-step method.
F we find In particular, because for the regardless of
, while for the when
F V V (43) , the coarse-step method will not show the expected
rise in nonlinear PMD in this limit. However, the poor
where V S S S S S S S treatment of nonlinear PMD in the coarse-step method has
S S S . Physically, the matrix V corresponds to the little practical significance for simulations of communications
backward transformation from to . By fibers to date in which both the actual nonlinear PMD and the
analogy with (16), we write nonlinear PMD generated by the coarse-step method are both
negligible.
V V (44) Our results suggest a modest improvement in the coarse-step
method. One could use an Euler angle transformation at the
(36)MARCUSE et al.: APPLICATION OF THE MANAKOV-PMD EQUATION 1745
end of each step which would guarantee complete mixing on REFERENCES
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PMD. We interpreted nonlinear PMD as the residue of
incomplete mixing of the Kerr nonlinearity over the Poincaré Dietrich Marcuse (M’58–F’73) was born in Koenigsberg, East Prussia,
Germany, in 1929. He received the degree of Diplom Physiker from the Freie
sphere. We next developed a numerical algorithm for solving Universitaet Berlin, Germany, in 1954 and the degree of Doktor Ingenieur
the Manakov-PMD equation and applied it to both NRZ from the Technische Hochschule, Karlsruhe, Germany, in 1962.
and soliton signals. We also examined the nonlinear PMD From 1954 to 1957, he worked at the Central Laboratory of Siemens and
Halske, Berlin, on transmission line problems and on the development of the
terms and showed that they are negligible even under fairly circular electric millimeter waveguide. From 1957 to 1994, he was a Member
extreme circumstances. Thus these terms can be safely of Technical Staff at AT&T Bell Laboratories, where he worked initially on
dropped in almost all cases, simplifying the numerical problems of wave propagation in circular electric millimeter waveguides and
on hydrogen cyanide maser. But throughout most of his career, he worked
algorithm. We next carried out an analysis of the widely on theoretical aspects of various laser problems and on the theory of wave
used coarse-step method, and we showed that it will yield propagation in optical fibers. From 1966 to 1967, he spent one year on
reasonable results in the parameter regime that is currently leave of absence from AT&T Bell Laboratories to teach at the Department of
Electrical Engineering of the University of Utah, Salt Lake City. Two one-
used in communication systems. At the same time, we month leaves were spent at the Institute of Advanced Studies of the Australian
note that the additional computation load required to solve National University, Canberra, in 1975 and 1990. After retiring from AT&T
the Manakov-PMD equation is slight, and it allows us Bell Laboratories, he became a part-time Visiting Research Professor at the
University of Maryland, Baltimore County. He has been a U.S. citizen since
to both accurately model particular representations of the 1966. He is the author of four books and 200 technical papers and holds 13
birefringence variation along the fiber and to determine the patents.
importance of the different physical effects that contribute Dr. Marcuse is a Fellow of the Optical Society of America. He received
the Quantum Electronics Award of the Quantum Electronics and Applications
to the Manakov-PMD equation in a physically transparent Society of the IEEE in 1981 and the Max Born Award of the Optical Society
way. of America in 1989.1746 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 15, NO. 9, SEPTEMBER 1997
C. R. Menyuk (SM’88) was born on March 26, P. K. A. Wai (SM’96) was born in Hong Kong
1954. He received the B.S. and M.S. degrees from on January 22, 1960. He received the B.S. degree
the Massachusetts Institute of Technology, Cam- from the University of Hong Kong, Hong Kong,
bridge, in 1976 and the Ph.D. degree from the in 1981 and M.S. and Ph.D. degrees from the
University of California, Los Angeles, in 1981. University of Maryland, College Park, in 1985 and
He had worked as a Research Associate at the 1988, respectively.
University of Maryland, College Park, and also In 1988, he joined the Science Applications In-
worked at the Science Applications International ternational Corporation, McLean, VA, as a Research
Corporation, McLean, VA. In 1986, he became an Scientist, where he worked on the Tethered Satellite
Associate Professor in the Department of Electrical Project. In 1990, he became a Research Associate
Engineering at the University of Maryland, Balti- in the Department of Electrical Engineering at the
more County, and was its founding member. In 1993, he became a Professor. University of Maryland, Baltimore County. In 1996, he became an Assistant
For the last few years, his primary research area has been theoretical and Professor at the Department of Electronic Engineering of the Hong Kong
computational studies of nonlinear and guided-wave optics. He has written Polytechnic University, Hong Kong. He was promoted to an Associate
computer codes to model the nonlinear propagation of light in optical fiber Professor in 1997. His research interests include theory of solitons, modeling
which have been used by industrial, governmental, and university research of fiber lasers, simulations of integrated optical devices, long distance fiber
groups throughout the United States. optic communication, and neural networks.
Dr. Menyuk is a member of the APS, OSA, and SIAM.You can also read
